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Questions tagged [contour-integration]

Questions on the evaluation of integrals along a locus in the complex plane.

This is the procedure of calculating the contour integral around a given path/contour. It allows us to evaluate integrals on the real line $\mathbb{R}$ that are not able to be evaluated using real-variable methods.

Links:

Contour Integration at Wolfram MathWorld

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Contour integral along a parabola

The question reads: Evaluate $$ \int_\gamma f(z)dz$$ where $$f(z)=x^2: x,y \in \mathbb{R} $$ and $\gamma$ is the parabola $y=2x^2$ from $x=0$ to $x=2$. This is the first question I've encountered where $f(z)$ is given in the form $u(x,y)$, and I've…
Forcefedglas
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Evaluate Contour Integral

I have provided my solution below, a confirmation on my solution would be appreciated, thanks in advance.
E.JJ
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Given $\vec{F}=(y,x)$, calculate contour integral $\int_C\vec{F}•\vec{dr}$, on length of closed path $C:\vec{r}(t)=(\cos t, \sin t), 0\leq t\leq2\pi$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 4 of 4, part $c$ and graded for 10%. Given $\vec{F}=(y,x)$, calculate the…
Dimitris S.
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Closed contour integral question with 3 line segments

I am really unsure as to how to tackle this contour integral question, Can I get a comprehensive guide to tackling this question? $$H(\lambda)= \oint_C \frac{e^{i\lambda z^2}}{z-2-i} \space dz$$ where the contour C comprises the straight line…
B DIll
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Contour integral with higher order poles on real axis

Is there any general solution of contour integral with higher order poles on real axis? I have got one but valid for simple pole…
Soumyajit Roy
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Locations of singularities of a function with respect to given contours

Show that $\int_{C_1}f=\int_{C_2}f$, where $C_1:|z|=1$, $C_2:|z|=2$, and $f(z)=\frac{2z+1}{\sin z}$. Hint: Locate the singularities of $f$ in each case and indicate their location with respect to the two given contours. I think I know how to…
Katie Shaw
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Branch cut jump of $\log z$

$$\int _ C \log z\,dz$$ where $C$ is a full circle in positive direction with radius $R$. I substitute $z=Re^{it}$, $dz=Rie^{it}dt$ $$\int _ 0 ^{2\pi} \log (Re^{it})Rie^{it}\,dt$$ $$\int _ 0 ^{2\pi} \log (R)Rie^{it}\,dt +\int _ 0 ^{2\pi}…
Gappy Hilmore
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Calculate contour integral

Calculate $\displaystyle \int_\Gamma \frac 1{z^4 + 81}$ where $\Gamma: |z+i| = \frac 34$ Can somebody help me with this question please or give me a hint on how to get started, as I have never seen a question with gamma like this and I have no…
bws
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Circle difference in contour integral

Let's say I am integrating a function over $|z| = 1$ and $|z-1| = 1$, is there any difference? I think the answer for both cases will be same, as in both cases, $$ z = \exp^{i\Theta} $$ and $$ dz = i \exp^{i\Theta} d\Theta $$ Please correct me if I…
simple
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Confusion regarding contour integral solution

In Schaum's complex variable book, there is an exercise in contour integration: $$ \int \overline{z}^{2} dz $$ over $|z|=1$. The answer seems to be $0$, but when I integrate like this using contour integration formula,…
simple
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Circular contour integration.

solving one of the 5 options would be much appreciated as this will give me an idea on how to solve the rest. Let $\gamma(w,R)$ denote the circular contour $t\mapsto w+Re^{it}$ where $0\lt t\lt2\pi$. Evaluate $$\int_\gamma\dfrac1{1+z^2}dz$$ when…
mike
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Contour Integral with Real Singularity

So, if I'm working in spherical coordinates, how would I evaluate the following integral? I know that I'm supposed to use contour integration and Jordan's lemma, but the fact that the singularity is located on the real axis is really throwing me…
Incognito
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Direction of Contour Integration

When I'm using the residue theorem to evaluate a contour integral, does the simply closed curve always have to be in a counter-clockwise direction? I believe that I can go in a clockwise direction, but this adds a negative sign, correct?
Incognito
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Logarithmic Contour Integration

So, I'm having a really difficult time trying to evaluate the following integral via contour integration (please, no other methods): $$\int_0^\infty{\frac{\log{(x^2+1)}}{1+x^2}} dx$$ Obviously, we're going to have branch cuts at z= +/- i which…
Incognito
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Contour Integration Part

I'm trying to evaluate the following integral, and I'm getting stuck on one part. Here's the integral: $$\int_{-\infty}^\infty \frac{\sin(x)}{x(x^2+1)} dx$$ Basically, I'm converting this to the complex plane and performing a contour integration…
Incognito
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